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Core Doppler Math: Modified Bernoulli, VTI, Stroke Volume & the Continuity Equation

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RDCS Adult Echocardiography AE

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1 Introduction & Exam Overview

The RDCS AE Exam, SPI Requirement & Logistics Content Outline, Domain Weights & How to Study

2 Cardiac Anatomy

Chambers, Walls, Myocardial Layers & the 17-Segment Model Cardiac Valves & Subvalvular Apparatus Great Vessels, Pericardium, Coronary & Venous Anatomy

3 Cardiac Physiology & Hemodynamics

Cardiac Cycle, Phases & Pressure-Volume Relationships Normal Systolic & Diastolic Function Hemodynamic Principles & Response to Stress

4 Echo Modalities, Instrumentation & Contrast

2D (B-Mode) and M-Mode Imaging Spectral, Color, and Tissue Doppler Modalities Image Optimization, Artifacts, and Contrast Echocardiography

5 Standard Views, Windows & Protocol

Parasternal Long- & Short-Axis Views Apical Views Subcostal & Suprasternal-Notch Views Exam Protocol, Patient Positioning & Provocative Maneuvers

6 Quantification & Doppler Hemodynamic Calculations

Chamber Quantification: LV, LA, RA, and RV Size, Mass, and Volumes Systolic Function: Ejection Fraction Methods, Fractional Shortening & Strain Diastolic Function Grading (ASE/EACVI 2016) Core Doppler Math: Modified Bernoulli, VTI, Stroke Volume & the Continuity Equation Advanced Hemodynamics: PISA/EROA, Pressure Half-Time, dP/dt, Qp/Qs & RVSP

7 Aortic & Mitral Valve Disease

Aortic Stenosis Aortic Regurgitation Mitral Stenosis Mitral Regurgitation

8 Right-Sided, Prosthetic & Infective Valve Disease

Tricuspid & Pulmonic Valve Disease and Pulmonary Hypertension Assessment Infective Endocarditis and Vegetations Prosthetic Valves and Postoperative Evaluation

9 Ischemic Disease & Cardiomyopathies

Coronary Anatomy, Ischemic Heart Disease & Wall-Motion Abnormalities Stress Echocardiography & Myocardial Viability Mechanical Complications of Myocardial Infarction Dilated & Restrictive Cardiomyopathy Hypertrophic Cardiomyopathy, SAM & LVOT Obstruction

10 Pericardial Disease, Masses & Aorta

Pericardial Effusion & Cardiac Tamponade Constrictive Pericarditis vs. Restrictive Physiology Cardiac Masses, Tumors & Thrombi Aortic Disease: Aneurysm, Dissection & Atheroma

11 Congenital Heart Disease & Shunts

Septal Defects & Shunt Quantification Common Adult Congenital Lesions

12 Clinical Care & Safety

Patient Care, Clinical-Data Integration & Exam Protocols Bioeffects, ALARA, Infection Control & Ergonomics

Core Doppler Math: Modified Bernoulli, VTI, Stroke Volume & the Continuity Equation

Key Takeaways Explain This Topic AI

The modified Bernoulli equation (ΔP = 4V²) converts a Doppler peak velocity in m/s directly into a pressure gradient in mmHg.

Velocity-time integral (VTI) is the area under the Doppler spectral trace and represents the stroke distance of blood at that site, measured in centimeters.

Stroke volume equals cross-sectional area times VTI (SV = CSA × VTI), and cardiac output equals stroke volume times heart rate (CO = SV × HR).

LVOT cross-sectional area is calculated as CSA = 0.785 × (LVOT diameter)², making the LVOT diameter measurement the largest source of error in stroke volume calculations because it is squared.

The continuity equation (AVA = CSA_LVOT × VTI_LVOT / VTI_AV) calculates aortic valve area from conservation of flow between the LVOT and the stenotic valve.

Last updated: July 2026

Why Doppler Math Matters

Measurement Techniques, Maneuvers, and Sonographic Views make up 25% of the AE content outline, and the calculations in this section are the mathematical foundation for nearly every quantitative hemodynamic assessment tested: pressure gradients, valve areas, stroke volume, and cardiac output. Hotspot and calculation-style items commonly present a traced Doppler envelope or a set of measured diameters and ask the candidate to apply one of these formulas directly, so accurate recall of each constant matters as much as conceptual understanding. Master these four relationships and the advanced calculations in Section 6.5, plus the valve-severity grading in Chapters 7-8, become straightforward substitutions into formulas you already know.

The Modified Bernoulli Equation

The full Bernoulli equation relates pressure and velocity across a flow convergence but includes terms for flow acceleration, viscous friction, and gravitational potential. In clinical echocardiography those terms are negligible for the high-velocity jets crossing stenotic or regurgitant orifices, so the equation simplifies to the modified Bernoulli equation :

ΔP = 4V²

where ΔP is the peak (or instantaneous) pressure gradient in mmHg and V is the peak (or instantaneous) jet velocity in m/s measured by CW Doppler. A 4 m/s aortic jet therefore corresponds to a 64 mmHg peak gradient (4 × 4² = 64 mmHg).

The simplified form assumes the proximal (upstream) velocity is low enough (<1 m/s) to be ignored. When the proximal velocity is elevated — a high-output state, a serial stenosis, or a small LVOT — the full modified equation must be used instead: ΔP = 4(V2² − V1²), where V2 is the distal (stenotic) velocity and V1 is the proximal velocity. Ignoring an elevated proximal velocity overestimates the gradient. Accurate gradients also require the Doppler beam to be as parallel to flow as possible — because velocity is squared in the equation, a beam-alignment error produces an even larger error in the calculated pressure gradient.

Velocity-Time Integral (VTI)

The velocity-time integral is the area traced under the spectral Doppler envelope for one cardiac cycle, measured in centimeters. Physically, VTI represents the "stroke distance" — how far a column of blood travels past the sample site with each beat. VTI is obtained by tracing the modal (outer-edge) velocity of the PW or CW spectral waveform and is the building block for stroke volume, cardiac output, the continuity equation, and — later in this guide — the dimensionless index and the Qp/Qs shunt ratio.

Stroke Volume and Cardiac Output

Because flow through a cylindrical column equals cross-sectional area times the distance the blood travels, stroke volume (SV) at any site is:

SV = CSA × VTI

For a circular structure such as the LVOT, cross-sectional area is calculated from the measured diameter (D, in cm):

CSA = π(D/2)² = 0.785 × D²

So the standard LVOT-based stroke volume is SV = 0.785 × (LVOT diameter)² × LVOT VTI , in mL. Multiplying stroke volume by heart rate gives cardiac output :

CO = SV × HR (mL/min, divided by 1000 for L/min)

Cardiac index (CI) further divides CO by body surface area. Typical adult resting values: LVOT diameter ≈1.8–2.2 cm, LVOT VTI ≈18–22 cm, SV ≈60–100 mL, CO ≈4–8 L/min, CI ≈2.5–4.0 L/min/m².

The Continuity Equation

The continuity equation applies conservation of mass to two sites in series along the same flow stream (no shunt or regurgitation between them): flow proximal to a stenotic valve equals flow through the stenotic orifice during the same ejection period, so CSA₁ × VTI₁ = CSA₂ × VTI₂ , or equivalently A₂ = A₁ × V₁ / V₂ using peak velocities in place of VTIs. Applied to aortic stenosis:

AVA = (CSA_LVOT × VTI_LVOT) / VTI_AV

using the LVOT as the known reference site and the aortic valve as the unknown orifice. The velocity-based shortcut (A₂ = A₁V₁/V₂) is faster but slightly less accurate since it ignores the shape of the flow envelope. A related, LVOT-diameter-independent shortcut is the dimensionless index (DVI) — VTI_LVOT ÷ VTI_AV, or equivalently V_LVOT ÷ V_AV — useful when the LVOT diameter is difficult to measure reliably; severity cutoffs for AVA and DVI are covered with aortic stenosis in Chapter 7.1.

Worked example: LVOT diameter 2.0 cm, LVOT VTI 24 cm, AV VTI 120 cm.

  • CSA_LVOT = 0.785 × 2.0² = 3.14 cm²
  • SV = 3.14 × 24 = 75.4 mL (near-normal)
  • AVA = (3.14 × 24) / 120 = 75.4 / 120 = 0.63 cm² — in the severe aortic stenosis range
  • Formula Reference Table
  • Calculation Formula Units
  • Modified Bernoulli ΔP = 4V² mmHg (V in m/s)
  • Full Bernoulli (elevated proximal V) ΔP = 4(V2² − V1²) mmHg
  • Cross-sectional area (circular) CSA = π(D/2)² = 0.785 × D² cm²
  • Stroke volume SV = CSA × VTI mL
  • Cardiac output CO = SV × HR L/min
  • Continuity equation (valve area) AVA = (CSA_LVOT × VTI_LVOT) / VTI_AV cm²
  • Dimensionless index DVI = VTI_LVOT / VTI_AV unitless

These relationships recur throughout the exam's Measurement Techniques domain and are prerequisites for the advanced hemodynamic calculations in Section 6.5.